On a non-singular projective variety, the complete linear system $|D|$ of a divisor $D$ is the set of all effective divisors linearly equivalent to $D$. Often we speak of the dimension $\mathrm{dim}|D|$, which is the number of parameters.

Consider in particular a complex projective variety, with an effective divisor $D$. Can there be disconnected components of $|D|$? For example, can it be that $\mathrm{dim}|D|=0$ while there are multiple elements in $|D|$?

Personally I would be particularly interested in examples of divisors $D$ on a complex surface where the self-intersection is negative, $D^2 < 0$: here $\mathrm{dim}|D|=0$ but I wonder if there may be several elements in $|D|$.

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    $\begingroup$ Remember that $|D|$ is a projective space, namely, $\Bbb P(H^0(\mathscr O(D)))$. $\endgroup$ – Ted Shifrin Mar 24 at 0:41
  • $\begingroup$ @TedShifrin Thanks a lot for your comment; this makes it totally clear. I will delete this question as I don't think it is useful for anyone else. $\endgroup$ – diracula Mar 24 at 0:49
  • $\begingroup$ Don't delete. I thought it was a good question. Perhaps I should turn my comment into an answer. $\endgroup$ – Ted Shifrin Mar 24 at 0:59

In fact, $|D|$ is a projective space — namely, $\Bbb P(H^0(D)))$. The linear system is coming from "continuous" variation of the divisor, as it consists of divisors that are linearly equivalent to the given divisor. Linear equivalence, in particular, is a specific sort of homotopy, and the divisors can be "connected" in a continuous way.


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